# Properties

 Degree 12 Conductor $2^{18} \cdot 3^{6} \cdot 5^{12}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − 6·3-s + 4-s + 2·8-s + 21·9-s − 6·12-s + 3·16-s − 12·24-s − 56·27-s − 12·31-s + 4·32-s + 21·36-s − 8·37-s − 20·41-s + 8·43-s − 18·48-s + 6·49-s − 12·53-s + 64-s − 24·67-s − 8·71-s + 42·72-s − 36·79-s + 126·81-s − 32·83-s + 28·89-s + 72·93-s − 24·96-s + ⋯
 L(s)  = 1 − 3.46·3-s + 1/2·4-s + 0.707·8-s + 7·9-s − 1.73·12-s + 3/4·16-s − 2.44·24-s − 10.7·27-s − 2.15·31-s + 0.707·32-s + 7/2·36-s − 1.31·37-s − 3.12·41-s + 1.21·43-s − 2.59·48-s + 6/7·49-s − 1.64·53-s + 1/8·64-s − 2.93·67-s − 0.949·71-s + 4.94·72-s − 4.05·79-s + 14·81-s − 3.51·83-s + 2.96·89-s + 7.46·93-s − 2.44·96-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 $$d$$ = $$12$$ $$N$$ = $$2^{18} \cdot 3^{6} \cdot 5^{12}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{600} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(12,\ 2^{18} \cdot 3^{6} \cdot 5^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )$$ $$L(1)$$ $$\approx$$ $$0.159819$$ $$L(\frac12)$$ $$\approx$$ $$0.159819$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5\}$,$$F_p(T)$$ is a polynomial of degree 12. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad2 $$1 - T^{2} - p T^{3} - p T^{4} + p^{3} T^{6}$$
3 $$( 1 + T )^{6}$$
5 $$1$$
good7 $$1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 47 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12}$$
11 $$1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12}$$
13 $$( 1 + 11 T^{2} - 16 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{2}$$
17 $$1 - 2 p T^{2} + 351 T^{4} - 1084 T^{6} + 351 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12}$$
19 $$1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12}$$
23 $$1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 4527 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12}$$
29 $$( 1 - 54 T^{2} + p^{2} T^{4} )^{3}$$
31 $$( 1 + 6 T + 77 T^{2} + 308 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
37 $$( 1 + 4 T + 51 T^{2} + 40 T^{3} + 51 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
41 $$( 1 + 10 T + 87 T^{2} + 588 T^{3} + 87 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
43 $$( 1 - 4 T + 65 T^{2} - 216 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
47 $$1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 7967 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12}$$
53 $$( 1 + 2 T + p T^{2} )^{6}$$
59 $$1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12}$$
61 $$1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12}$$
67 $$( 1 + 4 T + p T^{2} )^{6}$$
71 $$( 1 + 4 T + 101 T^{2} + 632 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
73 $$( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3}$$
79 $$( 1 + 18 T + 317 T^{2} + 2908 T^{3} + 317 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
83 $$( 1 + 16 T + 265 T^{2} + 2400 T^{3} + 265 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
89 $$( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2}$$
97 $$1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 24143 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Imaginary part of the first few zeros on the critical line

−5.62127773430141346814980798220, −5.57605520652319942218035698053, −5.55728717117692810265656600414, −5.50911098886993741796080005905, −4.89450136833995656433304553409, −4.89163429170202900306328328012, −4.77537189873276606493055123896, −4.63920355519819174369012170356, −4.51384057681432099411117707924, −4.18420043824646641370016337398, −4.05350888111005769883656762104, −3.93153066843160318929114544702, −3.61138327812209258684653296657, −3.33983476215168578230037569483, −3.29705382881241156709347973515, −2.86838385467434311690789796059, −2.84517853161744640264327861742, −2.37782268559277801117909772703, −2.02945077535826425175472285014, −1.59766349238880785080994304301, −1.59591393022744840411065286057, −1.41435310844905832860546651901, −1.33378189332377377405769252178, −0.54388433427110706472065004062, −0.15903604181634495935897673885, 0.15903604181634495935897673885, 0.54388433427110706472065004062, 1.33378189332377377405769252178, 1.41435310844905832860546651901, 1.59591393022744840411065286057, 1.59766349238880785080994304301, 2.02945077535826425175472285014, 2.37782268559277801117909772703, 2.84517853161744640264327861742, 2.86838385467434311690789796059, 3.29705382881241156709347973515, 3.33983476215168578230037569483, 3.61138327812209258684653296657, 3.93153066843160318929114544702, 4.05350888111005769883656762104, 4.18420043824646641370016337398, 4.51384057681432099411117707924, 4.63920355519819174369012170356, 4.77537189873276606493055123896, 4.89163429170202900306328328012, 4.89450136833995656433304553409, 5.50911098886993741796080005905, 5.55728717117692810265656600414, 5.57605520652319942218035698053, 5.62127773430141346814980798220

## Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.